click to select points on the graph. how many solutions does the system of equations have?
click to select points on the graph. how many solutions does the system of equations have?
Answer
Explanation:
Step1: Analyze the nature of the system of equations
A system of linear equations (y = m_1x + b_1) and (y = m_2x + b_2) has one solution if (m_1\neq m_2), no solution if (m_1 = m_2) and (b_1\neq b_2), and infinitely many solutions if (m_1 = m_2) and (b_1 = b_2). First, rewrite the equation (3x - 4y=-12) in slope - intercept form (y=mx + b). We have (3x-4y=-12), then (-4y=-3x - 12), and (y=\frac{3}{4}x + 3). The other equation is (y =-\frac{1}{2}x + 8).
Step2: Compare the slopes
The slope of the first equation (after rewriting) (m_1=\frac{3}{4}), and the slope of the second equation (m_2 =-\frac{1}{2}). Since (\frac{3}{4}\neq-\frac{1}{2}), the two lines intersect at exactly one point.
Answer:
The system of equations has one solution.